M-type estimator (<code>'l2'</code>, <code>'l1'</code>, <code>'huber'</code>,
or <code>'tukey'</code>).

estimator

A positive multiplier, or a vector of them, of \(\sigma\), the
square root of the variance, used in the cutoff parameter for the
<code>'huber'</code> and <code>'tukey'</code> estimators; should be <code>NULL</code> for the
<code>'l2'</code> or <code>'l1'</code> estimators.

Approximate asymptotic relative efficiency (ARE) of an M-type estimator to
the least-squares estimator given Gaussian errors, calculated using a tangent
space approximation.

Provides a gradient descent algorithm to find a geodesic relationship between real-valued independent variables and a manifold-valued dependent variable (i.e. geodesic regression). Available manifolds are Euclidean space, the sphere, hyperbolic space, and Kendall's 2-dimensional shape space. Besides the standard least-squares loss, the least absolute deviations, Huber, and Tukey biweight loss functions can also be used to perform robust geodesic regression. Functions to help choose appropriate cutoff parameters to maintain high efficiency for the Huber and Tukey biweight estimators are included, as are functions for generating random tangent vectors from the Riemannian normal distributions on the sphere and hyperbolic space. The n-sphere is a n-dimensional manifold: we represent it as a sphere of radius 1 and center 0 embedded in (n+1)-dimensional space. Using the hyperboloid model of hyperbolic space, n-dimensional hyperbolic space is embedded in (n+1)-dimensional Minkowski space as the upper sheet of a hyperboloid of two sheets. Kendall's 2D shape space with K landmarks is of real dimension 2K-4; preshapes are represented as complex K-vectors with mean 0 and magnitude 1. Details are described in Shin, H.-Y. and Oh, H.-S. (2020) <arXiv:2007.04518>. Also see Fletcher, P. T. (2013) <doi:10.1007/s11263-012-0591-y>.

are: Approximate ARE of an M-type estimator to the least-squares estimator

Description

Usage

Arguments

Value

References

See Also

Examples